= Solution
The <dimension of level-one cusp forms> is zero when $k$ is odd or $k<12$. For even $k\geq12$ it is
$$
\dim S_k(\Gamma(1))=
\begin{cases}
\left\lfloor k/12\right\rfloor-1,&k\equiv2\pmod{12},\\
\left\lfloor k/12\right\rfloor,&k\not\equiv2\pmod{12}.
\end{cases}
$$
Equivalently, multiplication by the <modular discriminant> gives $S_k(\Gamma(1))=\Delta M_{k-12}(\Gamma(1))$, and the displayed formula follows from the standard dimension formula for $M_{k-12}(\Gamma(1))$.
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